Question-related-to-Q-33217-If-A-1-A-2-A-n-are-n-points-with-integer-coordinates-of-a-plane-such-that-every-triangle-whose-vertices-are-any-three-of-the-above-points-has-its-centroid-with-at-lea

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Question Number 33323 by Rasheed.Sindhi last updated on 14/Apr/18

$$\text{You can't use 'macro parameter character \#' in math mode}$$$$\mathrm{If}{\mathrm{A}}_1,{\mathrm{A}}_2,\dots {\mathrm{A}}_{\mathrm{n}}\mathrm{are}\mathrm{n}\mathrm{points}\mathrm{with}\mathrm{integer}$$$$\mathrm{coordinates}\mathrm{of}\mathrm{a}\mathrm{plane}\mathrm{such}\mathrm{that}\mathrm{every}\mathrm{triangle}$$$$\mathrm{whose}\mathrm{vertices}\mathrm{are}\mathrm{any}\mathrm{three}\mathrm{of}\mathrm{the}\mathrm{above}$$$$\mathrm{points}\mathrm{has}\mathrm{its}\mathrm{centroid}\mathrm{with}\mathrm{at}\mathrm{least}\mathrm{one}$$$$\mathrm{non}-\mathrm{integer}\mathrm{coordinate}.\mathrm{Find}\mathrm{the}\mathrm{maximum}$$$$\mathrm{possible}\mathrm{n}.$$$$\mathrm{Recall}\mathrm{that}\mathrm{if}\mathrm{P}({\mathrm{x}}_1,{\mathrm{y}}_1),\mathrm{Q}({\mathrm{x}}_2,{\mathrm{y}}_2),\mathrm{R}({\mathrm{x}}_3,{\mathrm{y}}_3)$$$$\mathrm{are}\mathrm{three}\mathrm{vertices}\mathrm{then}\mathrm{centroid}\mathrm{G}\mathrm{is}$$$$(\frac{{\mathrm{x}}_1+{\mathrm{x}}_2+{\mathrm{x}}_3}{3},\frac{{\mathrm{y}}_1+{\mathrm{y}}_2+{\mathrm{y}}_3}{3})$$

Commented by Rasheed.Sindhi last updated on 17/Apr/18

$$\mathrm{n}=972$$$$\mathrm{There}\mathrm{sbould}\mathrm{be}\mathrm{at}\mathrm{most}972\mathrm{points}$$$$\mathrm{in}\mathrm{order}\mathrm{to}\mathrm{meet}\mathrm{the}\mathrm{condition}.$$

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